Do I need to fully charge my new phone before using it?

No... modern smartphone batteries are 'ready to go'... No need to 'fully charge first' - in fact: it's pretty much impossible to 'charge first' before use. When you plug an iPhone, a BB or most Android and WP handsets into the charger, the device turns on automatically! Thus it's in use.

Having said that, the battery will need a few charge cycles before it reaches its true potential. Expect disappointing battery performance the first few days. That is normal. After the normal phase, get angry